Question

if sides of a triangle are in ratio 3:4:5 and its perimeter is 36 cm, find its area. *

72 sq cm

32 sq cm

54 sq cm

67 sq cm​

Answer 1

Let the side of the triangle be 3x ,4x,5x

Now,

Sum of all sides = Perimeter

$3x + 4x + 5x = 36 \\ \implies 12x = 36 \\ \implies \: x = \frac{36}{12} \\ \implies \: x = 3$

So , the sides of triangle are

3*3=9cm

4*3=12cm

5*3=15cm

Now,

area of triangle :

AC^2=AB^2+BC^2

$\displaystyle{{9}^{2}+{12}^{2}={15}^{2}}$

Thus,It is a right angle triangle

Area of triangle abc :

$area \: of \: triangle \: abc \: = \frac{1}{2} \times base \times height$

$\implies \frac{1}{2} \times 9 \times 12 \\ \implies \frac{1}{ \cancel 2} \times 9 \times \cancel12 \\ \implies54 {cm}^{2}$

Answer 2

Given :-

Ratio of the side of a triangle = 3 : 4 : 5

Perimeter of the triangle = 36 cm

Required to find :-

• Area of the triangle

Formula used :-

Heron's formula

$\green{ \boxed{ \text{ \pink{area } \blue{= }} \tt{ \red{ \sqrt{s(s - a)(s - b)(s - c)} }}}}$

Solution :-

Given information :-

The sides of a triangle are in the ratio of 3 : 4 : 5

Perimeter of the triangle = 36 cm

So,

Let the side of the triangles be ; 3x , 4x & 5x

We know that ;

Perimeter of the triangle = Sum of all its sides

=> 36 = 3x + 4x + 5x

=> 36 = 12x

=> 12x = 36

=> x = 36/12

=> x = 3

Hence,

The sides of the triangle are ;

• 3x = 3 ( 3 ) = 9 cm

• 4x = 4 ( 3 ) = 12 cm

• 5x = 5 ( 3 ) = 15 cm

Since we don't know the measurement of the height ( Altitude ) . we need to use the Heron's formula .

Heron's Formula

$\tt{ \pink{ area = \sqrt{s(s - a)(s - b)(s - c)} }}$

Here,

s = semi - perimeter

a , b , c = The three sides of the triangle

Let's find the semi - perimeter ;

The word " semi " means " half " . So , semi- perimeter is half the perimeter .

This implies,

Semi - perimeter ( s ) = 36/2

S = 12 cm

Substituting the values in the formula ;

$\tt{area = \sqrt{18(18 - 9)(18 - 12)(18 - 15)}}$

$\tt{area = \sqrt{18(9)(6)(3)} }$

$\tt{area = \sqrt{ 18 \times 9 \times 6 \times 3}} \\ \\ \tt{area = \sqrt{ 6 \times 3 \times 3 \times 6 \times 3 \times 3}} \\ \\ \tt{area = \sqrt{ {6}^{2} \times {3}^{2} \times {3}^{2} } } \\ \\ \tt{area = \sqrt{ {6}^{2} } \times \sqrt{ {3}^{2} } \times \sqrt{ {3}^{2} } } \\ \\ \tt{area = 6 \times 3 \times 3} \\ \\ \tt {area = 54 \: {cm}^{2} }$

Therefore,

Area of the triangle = 54 cm²